Potential energy surface

A potential energy surface maps a system’s potential energy as a function of its nuclear positions. Its shape describes stable structures, reaction pathways, and energy barriers.

Potential energy: Energy associated with the configuration of a system, often represented as a function of position. A potential energy surface assigns this energy to each nuclear configuration.
Potential energy gradient: The vector of derivatives of potential energy with respect to coordinates. Its negative gives the force driving nuclei downhill across the surface.
Transition state theory: A theory that estimates reaction rates from the properties of a dividing surface near the transition state. It uses activation barriers and transition-state properties derived from energy surfaces.
Free energy surface: A map of a system’s free energy over selected coordinates, incorporating entropic contributions under specified conditions. Unlike a potential energy surface, it includes thermodynamic effects from degrees of freedom that may be omitted.
Conical intersection: A degeneracy between electronic potential energy surfaces forming a seam in nuclear-coordinate space. These intersections create regions where separate smooth single-state descriptions break down.
Molecular geometry: The three-dimensional arrangement of atoms in a molecule. A geometry supplies the nuclear positions used as coordinates on the surface.
Hessian matrix: A square matrix of second derivatives of a function with respect to its variables. Its eigenvalues reveal local curvature and distinguish minima from saddle points.
Quantum chemistry: The application of quantum mechanics to calculate the structures and properties of atoms and molecules. Electronic-structure calculations provide energies across nuclear configurations.
Potential of mean force: An effective free-energy profile obtained by averaging over degrees of freedom not explicitly represented. It is a statistical, coordinate-dependent landscape rather than the bare potential energy map.
Avoided crossing: A region where two interacting energy levels approach but do not become degenerate as a parameter changes. It can sharply reshape surfaces and enable transitions between electronic states.

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